TBs are bred to have no bottom…when they get exhausted, they find another gear. Their training does not include much canter, if any. They gallop, run and most have a good long trot if they relax a little.
IMO the biggest mistake some make with OTTBs is ignoring the fact not only do they not understand how, they lack the correct muscles. Trying to force it leads to making a real mess. They physically can’t do it and it can get them mentally lit up or sore trying to force it, or they run out of the ring with a rider not understanding why that happened.
OP says this one has a good, long walk so that’s what I would focus on to start rebuilding the muscling need for riding horse work. Wouldn’t canter for now, consistent daily work focusing on relaxation and learning to relax the body into long and low and the walk gives them a place to got to when they get excited, loose rein walk. Add in circles, changes of direction and halt. When they get that, you can add some trot, drop back to walk if they get excited. Add canter a little at a time. Each time you add a gait, do changes of direction and circles. Crisp transitions are a long way away, don’t force those or you will either back them off or light the fuse. Or both, keep them busy with what they know and go back to what they know even if it’s a boring walk. Patience.
If this one is not out 24/7, you will have some excess energy problems which can lead to lack of focus. Probably going to need to make sure the horse is ridden and schooled CORRECTLY at least 4-5 days a week, they are doing something daily at the track, they are bred to be worker bees and need a job. Likely lunging is going to be in the picture but turn that into a meaningful lesson, they need to be taught how, they don’t lunge at the track, they get on them. Trying to do that in a big area is asking for trouble, round pen or a small paddock or corral is best. They need to wear a bridle and surcingle/saddle to define this as a work situation, not buck and fart playtime.
I like to use properly adjusted side reins on developing horses, correctly used they not only give some control to the handler, they can teach the horse to carry itself and develop some muscle memory to do so along with the muscles themselves. Start at the walk and trot though, no forcing and the idea is to prepare them for riding, not replace it. IMO, many off the track have seen sidereins or similar before, last few times I’ve seen morning conditioning workouts, most have been in some additional headgear.
Unfortunately the second biggest mistake some of those those who buy OTTBs without any knowledgeable transition training make is not realizing the time it takes to develop a consistent program most days of the week and stick to it. Closely followed by not being skilled enough to administer that program or afford a regular rider. That’s bad for the horse who is often labeled difficult or crazy, they get sold along to less then ideal situations simply because of human buying mistakes.
If OP is only working with this one once or twice a week, it’s going to be really difficult. And if the owner rides it the other days, it’s going out arena gate with her again. The owner needs the same lessons OP gives the horse and needs to do the homework every time every ride or it won’t stick.
I wasted a lot of time with situations like this way back when I dabbled at free lancing. Like beating your head against the wall sometimes.